Modular forms in Quantum Field Theory
Francis Brown, Oliver Schnetz
TL;DR
The paper investigates the arithmetic structure of perturbative quantum field theory by studying point-counts over finite fields of graph hypersurfaces associated with Feynman graphs. It reports an experimental analysis of counts modulo $q^3$ for graphs up to 10 loops, finding many instances where the counts match Fourier coefficients of modular forms with weights $\leq 8$ and levels $\leq 17$. The work connects these observations to the $c_2$-invariant and graph periods within the $\phi^4$-theory framework, highlighting the role of graph polynomials and primitive-divergent subgraphs. These results suggest a deeper modular structure in perturbative QFT amplitudes and motivate further number-theoretic explorations of graph hypersurfaces.
Abstract
The amplitude of a Feynman graph in Quantum Field Theory is related to the point-count over finite fields of the corresponding graph hypersurface. This article reports on an experimental study of point counts over F_q modulo q^3, for graphs up to loop order 10. It is found that many of them are given by Fourier coefficients of modular forms of weights <=8 and levels <=17.